REVIEW OF
BASIC MATHEMATICAL CONCEPTS
The following document is a modified version of a
laboratory exercise prepared for students taking
general biology. It is provided here for individual
review purposes only and may not be used for
any other purpose or in any other context.
J. Montvilo
Review of Basic Mathematical Concepts
Version prepared 8/23/03; for individual review purposes only.
Page 1
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Review of Basic Mathematical Concepts
Version prepared 8/23/03; for individual review purposes only
REVIEW OF BASIC MATHEMATICAL CONCEPTS
Jerome A. Montvilo
Copyright © 2003. All rights reserved.
Mathematics is the door and the key to the sciences.
— Roger Bacon
1. INTRODUCTION
s Throughout high school, and perhaps even before and after, you may have
sat in class after class of arithmetic, algebra, geometry, trigonometry and the
rest asking yourself, “When am I ever going to really use this stuff, anyway?”
The answer is: now.
There was a reason for sitting through all those classes. As with this biology
course, it was and is to prepare you for the future, since you can’t always
know or predict what and when it really would be useful to know these
things1. Since it is possible that you may not have paid attention in those
classes as well as you might have, or because even if you did learn the
material you may have forgotten it through lack of use, the basics of scientific
mathematics will be reviewed here.
s Like it or not, science is based on numbers. When data are2 collected they
must be analyzed to glean the information from them. This often means
statistical analysis of some sort. Once the information is obtained, it must be
presented clearly. This is often accomplished most effectively in the form of
graphs. Statistical analysis of data will be covered in several labs in the future,
so it will not be covered here. Because of its importance to presenting and
analyzing the data you obtain in other labs in this course the basic concepts of
graph preparation will be reviewed below.
s It is not possible here to review all of the mathematical concepts that
should have been learned previously. What will be attempted is to provide you
with enough information to be able to solve the basic mathematical and
statistical problems you are likely to encounter in this and other similar
courses.
Some, if not all, of what follows below may be familiar to you. The purpose
of putting it here is to make it familiar to everyone in the course or as a
review. Even if you don’t think you need to go over this material, do so to
1
It is also why this author, a biologist who never did particularly well in mathematics, is now writing an
introduction to basic mathematical concepts. Go figure (no pun intended).
2
The word “data” is plural, so technically the correct usage is “data are”. (The singular of “data” is
“datum”.) However, in the recent past it has become more common, and acceptable, to use the word “data”
as a singular noun. This newer convention will not be followed here.
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refresh your memory. If you don’t remember this material, or if you have
never had it before, take your time and try to learn it as best you can.
s There is one important thing you can do to make sure that your
mathematical calculations are correct. Simply ask yourself, “Does this look
right? Does it make sense?” If you multiply 5 times 10 you should get 50.
Therefore, if you multiply 5 times 9.9 you should get something near 50. If
your answer does not seem to be in the same range as your estimate check
your results again. (Assuming, of course, your estimate was correct in the first
place, but that’s another problem!)
2. SOME FUNDAMENTAL IDEAS AND RULES
s Much of what follows should go without saying, but sometimes the
fundamentals are forgotten. They are reviewed here for your convenience.
Numbers are written by using the symbols 0 1 2 3 4 5 6 7 8 9. Except for 0
(zero) they all have a specified value. Zero signifies no value. For an unknown
value an “x” is often used and for an unspecified value an “n” may be used.
l
Numbers may be manipulated (operated on) in various ways, and symbols
are used to indicate what operations are to be carried out. These operations
(and symbols) include the basic ones such as addition (n + n), subtraction (n –
n), multiplication (n ¥ n or n * n or n • n or n(n)), and division (n ÷ n or n ⁄ n).
l
Numbers also may be manipulated by squaring, by cubing, or by being
raised to some other power [indicating that the number is multiplied by itself a
certain number of times] (n2, n3, nn). (The superscript number is called the
exponent; on some calculators it may show up as “En”, as in 10E2 = 102, or in
some other way.) Numbers may be manipulated by taking the square root,
cube root, or some other root [essentially the reverse of squaring, cubing, and
so on] ( , 3 , n ).
l
Numbers may be compared to each other and may found to be equal (=) or
approximately equal (ª) to each other. One number may be greater than (>)
the other number or less than (<) the other number.
l
Other more complex manipulations and comparisons may also be carried out,
but they will not be covered here.
Numbers less than zero are said to be negative (–) numbers. The larger a
negative number is the lower its value (for example, –10 < –5). Numbers
greater than zero are said to be positive (+, but usually implied so not written)
numbers. These numbers may extend to infinity (•) in either direction.
l
These relationships may be seen most easily on a “number line”, as shown
below, where the values to the left are always less than values to the right.
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… –100… –50… –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 …50 …100 …
When the sign (negative or positive character) of a number is not important
for a calculation but only its numerical value matters then the absolute
(positive) value (n) is used.
l
When two positive numbers are added together the result is a higher positive
number. When two negative numbers are added together the result is a lower
negative number. Whether the result of an addition or subtraction calculation
involving positive and negative numbers is positive or negative depends on
the magnitude of the values of the numbers involved.
l
Two negative numbers multiplied together results in a positive number, as
do two positive numbers multiplied together. A positive number multiplied by
a negative number results in a negative number.
l
A negative number divided by a negative number results in a positive
number. A negative number divided by a positive number results in a negative
number. A positive number divided by a negative number results in a negative
number. A positive number divided by a positive number results in a positive
number.
l
Any number, negative or positive, divided by itself results in a value of 1
(positive 1).
l
l
A number divided by 1 equals itself.
Manipulations involving zero are special. Any number multiplied by zero
equals zero. You may not divide by zero.
l
Fractions will not be covered in detail here. However, fractions can easily be
converted into decimals, the preferred method of dealing with values less than
1, by dividing the numerator (on top) buy the denominator (on the bottom).
(The “⁄” symbol can be thought of as meaning “divided by”.) For example, 1⁄10
= 0.1 and 5⁄8 = 0.625. This is most easily accomplished using a calculator.
l
Units, such as degrees ( °, °F, °C ), length units (m, mm, cm, etc.), area units
(mm2, cm2, etc.), and so on, must be included in the reporting of numbers. (In
a table where the units are clearly given in the label the units may be left out
to reduce clutter.)
l
s Other more detailed concepts will be covered in the sections below.
3. A COUNTING SYSTEM
s Our decimal counting and mathematical systems are founded on the
concept of base 10 arithmetic. (“Deci” refers to “tens”.) Put simply, this
means that the position of a numeral in a sequence of numerals will determine
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its value. This is so because the numeral indicates the number of “powers of
ten” that are found in that position. To put it another way, that numeral is
multiplied by the power of ten determined by the place it is in. Zero is used as
a “placeholder” only and has no value of its own.
The positions begin as follows and continue in both directions from the
decimal point, as in the chart below:
thousands
1000s
hundreds
100s
tens
10s
ones
1s
•
tenths
.1s
hundredths
.01s
thousandths
.001s
(The decimal point, of course, indicates the separation between the whole
numbers on the left and the fractional numbers on the right.)
This is basically a multiplication and addition scheme. For example, the
number 2301.235 could be represented in the above chart as follows:
thousands
1000s
2
hundreds
100s
3
tens
10s
0
ones
1s
1
•
.
tenths
.1s
2
hundredths
.01s
3
thousandths
.001s
5
This means that you have 2 thousands (2 ¥ 1000) added to 3 hundreds (3 ¥
100) added to 0 tens (0 ¥ 10) added to 1 one (1 ¥ 1) plus 2 tenths (2 ¥ 1⁄10)
added to 3 hundredths (3 ¥ 1⁄100) added to 5 thousandths (5 ¥ 1⁄1000). Shown
another way, it might look something like this:
2 thousands
3 hundreds
0 tens
1 one
2 tenths
3 hundredths
5 thousandths
total:
2000.
300.
00.
1.
.2
.03
.005
2301.235
There are several points of significance to this scheme, not the least of which
is that it can be used to simplify mathematical manipulations through the use
of so-called “scientific notation” which will be discussed below.
4. SCIENTIFIC NOTATION
s You may have noticed that each of the numerical positions discussed
above was either ten times greater than or ten times less than the position on
either side of it. Therefore, for example, the position signifying “tens” is ten
times larger than the “ones” position but only one-tenth as large as the
“hundreds” position. Why is this useful?
As noted above, each position can be indicated as a “power of ten”. The
power to which ten is raised is indicated by a superscript number (exponent)
after the 10. Ten is multiplied by 1 to give you 10. In other words, 101 = 10.
Ten is multiplied by 10 to give you 100, or 102 = 100. Each of the positions
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can be indicated by a particular power of ten (how many tens are multiplied
together), as shown below:
thousands
1000s
3
10
hundreds
100s
2
10
tens
10s
1
10
ones
1s
0
10
•
tenths
.1s
10–1
hundredths
.01s
10–2
thousandths
.001s
10–3
Notice that 100 equals 1. Notice that as you move to the left of the decimal
point the power of ten is raised by an additional +1 for each position. Notice
that as you move to the right of the decimal point the power of ten is raised by
an additional –1 for each position. Notice that the numeral in the exponent
indicates how may tens away from the decimal point the number is and the
sign indicates whether it is to the left of the decimal point (+, larger value) or
to the right of the decimal point (–, lower value). Why is this important?
For one thing, it provides an easy way to do complicated problems.
Multiplication problems are converted into addition problems and division
problems are converted into subtraction problems. How?
Notice that if you multiply 10 times 10 you get 100. Or, 101 ¥ 101 = 102. 100
times 10 equals 1000 (or 102 ¥ 101 = 103). To multiply these two numbers,
then, you just add the exponents.
What is 1000 divided by 10? 100. [103 ⁄ 101 = 102. (1000⁄10 = 100).] Just
subtract the exponents.
What if you want to add or subtract numbers using exponents? Then it
becomes a little tricky. You can only do this if the exponents are the same.
You cannot directly add 103 plus 101, for example. You first need to convert
one of the numbers so that the exponents are the same. How do you do this?
Usually, by moving the decimal point which, in effect, changes the exponent.
Note that in the examples above only the base number and the exponent were
given. It was assumed that you only had one of them. To be more correct,
these numbers should have been written as something like 1 ¥ 103 or, more
correctly, 1.000 ¥ 103.
Notice that if you move the decimal point to the right three spaces you will
have a value of 1000. That’s what the exponent means.
Every time you move a decimal point to the left one place the exponent loses
a value of one. Every time you move a decimal point to the right the exponent
gains a value of one. So, the number 1000 could be written as 1.000 ¥ 103 (1 ¥
1000) or as 10.00 ¥ 102 (10 ¥ 100) or as 100.0 ¥ 101 (100 ¥ 10) or as 1000. ¥
100 (1000 ¥ 1). The number .01 could be written as 1.00 ¥ 10–2 (1 ¥ .01) or as
10.0 ¥ 10–3 (10 ¥ .001) and so on. (While it could be done, it is not an
accepted method of writing numbers in scientific notation. It is used here just
as an example.)
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What about a number such as 106? Can it be written using an exponent? The
answer is yes. “106” simply means that you have one 100 and six 1s. 106,
then, means 100 (1 ¥ 102) plus 6 (6 ¥ 100). But remember, you cannot add or
subtract these kinds of numbers unless the exponents are the same. Therefore,
either the 102 or the 100 must be changed. Which one? Technically, it doesn’t
matter, but in this case it is easier to change the 100. The decimal point needs
to be shifted two spaces to the left. Since 6. ¥ 100 is the same as .6 ¥ 101
which is the same as .06 ¥ 102, it can now be seen that 106 can be written as
the sum of 1 ¥ 102 plus .06 ¥ 102, or 1.06 ¥ 102.
(Note that for consistency additional zeros were not added to the decimal
numbers above. Technically, and from here on, so there is no ambiguity, a
leading zero should and will be added in front of the decimal place. Therefore,
.1 will be written as 0.1 and so on. Adding a zero after the decimal point, as in
changing 1 to 1.0 should not and will not be done because to do so changes
the accuracy and precision of the number, a matter taken up in the discussion
of significant figures.)
These same rules apply to all numbers and form the basis of what is known as
scientific notation.
The primary usefulness of scientific notation lies in the fact that both very
large and very small numbers can be written in a compact manner. The
number 10000000000 can be written as 1 ¥ 1010. (Notice that the positive
exponent here indicates the number of zeros—representing powers of 10—in
the number.) The number 0.000001 can be written as 1 ¥ 10–6. (Notice that the
negative exponent here indicates how many places there are to the right of the
decimal point.)
The main difference between what was discussed at the beginning of this
section and scientific notation is that it is most common to express a number
in scientific notation as a single digit from 1 to 9 followed by a decimal point
and the appropriate rest of the decimal followed by the exponential power of
ten to which that decimal is raised. Therefore, the number 365.249 is
expressed in scientific notation as 3.65249 ¥ 102. (Moving the decimal point
two spaces to the right returns the decimal equivalent.)
Briefly, then, to convert a decimal number to its scientific notation equivalent
you move the decimal so that a single digit appears before the decimal point.
If you moved the decimal point two places to the left then the exponent is a
positive 2. If you moved the decimal point two spaces to the right then the
exponent is a negative 2. To convert a number from scientific notation to its
decimal equivalent you would just reverse the process. A positive exponent
indicates to move the decimal point from where it is to the right the number of
places designated by the exponent. A negative exponent indicates to move the
decimal point from where it is to the left the number of places designated by
the exponent.
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These manipulations are summarized in the table below.
FROM DECIMAL NOTATION
TO SCIENTIFIC NOTATION
FROM SCIENTIFIC NOTATION
TO DECIMAL NOTATION
1359.234 Æ 1.359234 ¥ 10
(The decimal point is moved 3 places to the left to
give a single digit before the decimal point, so the
3
exponent is +3. As a check, remember that 10 =
1000, so this looks right.)
0.000149 Æ 1.49 ¥ 10–4
(The decimal point is moved 4 places to the right
to give a single digit before the decimal point, so
the exponent is –4. As a check, remember that
10–4 represents 0.0001, so this looks right.)
3.458 ¥ 10 Æ 345.8
(The +2 exponent indicates that the decimal point
should be moved 2 places to the right. As a
2
check, remember that 10 represents 100 and
therefore 3.4 times 100 would be about 340.)
6.385 ¥ 10–3 Æ 0.006385
(The –3 exponent indicates that the decimal point
should be moved 3 places to the left; zeros are
added as necessary. As a check, remember that
10–3 represents 0.001 and 6 times 0.001 would be
0.006.)
3
¥
4.1
2
Convert each of the following numbers either from decimal notation to
scientific notation or from scientific notation to decimal notation. Place your
answers on the worksheet.
4.1a
4.1b
4.1c
1,359 Æ ?
45.79 Æ ?
0.0017 Æ ?
4.1d
4.1e
4.1f
2.3 ¥ 103 Æ ?
1.356 ¥ 102 Æ ?
4.76 ¥ 10–3 Æ ?
5. ORDER OF OPERATIONS
s Most non-arithmetic errors in calculations are the result of not following
the correct “order of operations” in the calculations. While this can become
problematical if an equation is not set up properly, in a well laid out equation
what the operations are should be clear and the results unambiguous.
By way of review, the operations to be dealt with here are the following:
raising to a power, multiplication, division, addition, and subtraction. These
should all be familiar at this point.
In order to “force” a calculation between two numbers they may be included
in parentheses, ( ), also sometimes called brackets. Liberal use of parentheses
simplifies the problem of what calculations to do first in an equation.
s Although there may be minor exceptions, the general rule is that operations
are carried out in the following order from left to right1:
[parentheses] [exponents] [multiplication and/or division] [addition and/or subtraction]
1
How do you remember the order? One old mnemonic (memory aid) is the following sentence:
Please
excuse my dear Aunt Sally. The first letter of each word corresponds to the first letter of an operation. The
operations are carried out in the same order as they occur in the sentence.
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(Actually, what is really meant is that any calculation or number that occurs
within a “grouping symbol” such as the parentheses, ( ), but also straight or
curly brackets, [ ] or { }, the absolute symbol (representing only the positive
value of the number is used),  , or the radical/root sign,
, should be
carried out or evaluated first. Then exponents should be evaluated. Then,
working from left to right, multiplications and divisions are done—you do the
divisions first if they appear first from left to right. Finally, additions and
subtractions are done—you do the subtractions first if they appear first from
left to right.)
NOTE: Do not confuse the slash symbol used to represent a division operation
(/) with the line separating the numerator from the denominator in a fraction
(⁄). While they look similar (but look carefully and you will see that are not
exactly the same) and mean basically the same thing, the interpretation in
terms of order of operation may be different. In a properly written equation
this should not be a problem. In the examples that follow the / symbol
5
represents a division, as in 10 / 5 = 2. Fractions will be written as 5⁄10 or
.
10
s What is the result of the following calculation?
5+6¥7=?
Is it 5 + 6 then multiply by 7 to give you 77? Or is it 6 x 7 then add 5 to give
you 47? Following the rules for the correct order of operations the answer is
47. The multiplication is done first (6 x 7) and then the addition (5 + 42).
s What is the result of the following calculation?
(5 + 6) ¥ 7 = ?
Now the answer is 77 because you do the operation within the parentheses
first and then do the multiplication.
s One final example follows:
(75 + (70 + 5)) + 4 + 12 ⁄ 6 – 3 ¥ 25
The expression in the parentheses is evaluated first. Where there are nested
parentheses, as here, always work from the inner parentheses outward.
Therefore, 70 + 5 = 75. 75 + 75 = 150. This leaves the equation as:
å
150 + 4 + 12 ⁄ 6 – 3 ¥ 25
†
ç
The exponent is now calculated. 25 = 32. This leaves the equation as:
150 + 4 + 12 ⁄ 6 – 3 ¥ 32
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Next, the multiplication and division are carried out from left to right. 12 ⁄ 6
= 2. 3 x 32 = 96. This leaves the equation as:
é
150 + 4 + 2 – 96
Finally, the additions and subtractions are done. These are normally done
from left to right, giving: 156 – 96 = 60.
è
60
s It is very important that these rules be followed carefully in order to get the
correct answers in your calculations.
¥
5.1
What are the results of the following calculations?
5.1a
5.1b
5.1c
((53 + 21) ⁄ 12) ¥ 3 =
4 ¥ 10 + 34 – 14 =
((|–3| ¥ 3) ⁄4) ¥ 5 – 2 =
5.1d
5.1e
5.1f
12 ¥ 0.5 ¥ 3 + 6 – 3 =
24 ⁄ 3 + 5 ¥ 3 =
(24 ⁄ (3 + 5))¥ 3 =
6. RATIOS AND PROPORTIONS
s A ratio is a comparison made between two or more numbers. In a situation
in which there is one of one type of thing and three of another type the ration
is said to be a three to one ratio. This is usually written in one of three ways:
1 to 3
1:3
1
1
⁄ 3 or
3
What this means, simply, is that for every one of the first thing there are three
of the second things.
Depending on what is being represented, it is also possible to indicate that for
every three of the first thing there is only one of the second things, as seen
below.
3 to 1
3:1
3
3
⁄ 1 or
1
In biology, especially in genetics, you can have a situation where you are
comparing the amounts of more than two things. For the traits of offspring, for
example, you may have nine of one trait (called a phenotype), three of another
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phenotype, three of another phenotype, and one of another phenotype giving
what is known as a 9 : 3 : 3 : 1 phenotypic ratio.
s A proportion is a comparison between two ratios. The comparison is
usually made between two ratios that have the same numerical relationships
so that they are usually separated from each other by an equal sign. An
example follows:
1 5
=
2 10
This means that each of the expressions on either side of the equal sign is
equivalent. This is convenient if you need to find out an unknown amount, as
in the example below. (As usual, an “x” stands for the unknown quantity.)
In a very simple case, let’s say that one horse can eat two bales of hay in one
day. How many bales of hay would five horses eat? Set up as a proportion the
relationship would look like this:
1 5
=
2 x
Notice that the same type of objects is either above or below the line in each
ratio:
1 horse
5 horses
=
2 bales of hay x bales of hay
This may be read as: 1 horse is to 2 bales of hay as 5 horses is to how many
bales of hay?
The answer is obtained through cross-multiplication and simple algebra. The
two numbers that are diagonal to each other in the relationship are multiplied
and the equivalence is kept. This gives you:
1O N5
2N O x
1¥ x = 2 ¥ 5
x=
2 ¥ 5 10
=
= 10
1
1
It would therefore take ten bales of hay to feed five horses.
¥
6.1
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Thirty-three red maple trees were found in a section of forest covering an area
two square kilometers in size. What is the probable number of red maple trees
present in the whole forest if the entire forest covers 98 square kilometers?
Review of Basic Mathematical Concepts
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s As can be seen from the above examples, in a typical proportional
relationship as one value goes up so does the other. In an inversely
proportional relationship the reverse is true—as one value goes up the other
goes down or as one value goes down the other goes up. Using the example
above, as you increase the number of horses the number of bales of hay
needed also increases. This is a proportional relationship. If it takes one
person one day to do a job, how long will it take two people? The answer is
one-half day. This is an inversely proportional relationship, since as the
number of people goes up the time it takes to complete the job goes down.
7. PERCENTAGES
s A percentage is merely a ratio in which one of the numbers is always 100.
(“Cent” refers to 100; “per” means “out of” or “for each”; “percent” therefore
means a number out of 100.) The symbol % represents “percent”. If you say
that something makes up 33 % of the total this means that for each 100 things
that there are 33 of them will be of that certain type.
This can actually be represented in three different ways, all of which are
equivalent:
33% =
33
= 0.33
100
(To convert the decimal to a percentage just multiply by 100 and add the
percent sign.)
Since it is often easiest to compare two quantities when they are both
represented using equivalent measurements, in this case percentages, it is
usually a simple matter to convert all data compared to percentages, as shown
in the example below.
s You receive a 67 out of 80 on one exam and a 47 out of 65 on another
exam. In terms of percentages, which grade is higher?
Clearly, the answer would be easy if both exams were scored out of 100. What
can be done is to use the techniques learned above to convert each of the
grades to a percentage, as if they actually had been graded out of 100.
For the first exam set up the following proportion:
67
x
=
80 100
Do the math as shown in the previous section (it does not matter which of the
cross-multiplied numbers goes first):
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80 ¥ x = 67 ¥ 100
x=
67 ¥100 6700
=
= 83.75
80
80
The results of the first exam, then, are equivalent to a grade of 83.75 out of
100.
For the second exam a similar process is gone through:
47
x
=
65 100
65 ¥ x = 47 ¥ 100
x=
47 ¥100 4700
=
= 72.31
65
65
The results of the second exam are equivalent to a grade of 72.31 out of 100.
In terms of percentage, then, you did better on the first exam than on the
second exam.
(A quick way to convert any fraction to a percentage, especially on a
calculator, is to do the division indicated by the fraction and then multiply the
result by 100.)
s There are other ways of doing these types of problems, but they all should
yield comparable results.
¥
7.1
A biologist collect 940 containers of pond water in order to study the single
celled organism called Paramecium. Of those containers, only 47 contained
Paramecium. What percentage of all the containers is represented by
Paramecium-containing containers?
8. SIGNIFICANT FIGURES AND DIGITS1
s Some of the trickiest basic mathematical concepts to grasp are the concepts
of significant figures (sometimes called significant digits or significant
numerals) and rounding. Although the idea behind these concepts is simple,
carrying them out is not. This is because there is not complete agreement on
how they are carried out.
1
There are a number of good tutorials on the World Wide Web that deal with mathematical concepts. The
one from which some of the following material comes can be found at the following URL:
http://ww.chemtutor.com/numbr.htm#meas
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The concept of significant figures is related to measurement, precision, and
accuracy, which will be covered in more detail in another lab exercise. (For
now it is important to know that “accuracy” refers to how close a number is to
the actual value and “precision” refers to the exactness of the number.) Simply
put, there are “exact numbers” and “measured numbers”.
s Exact numbers are exact because they are defined that way. By definition,
for example, there are 2.54 centimeters in 1 inch and there are 1.8 Fahrenheit
degrees in 1 Celsius degree. A number may also be exact because the numbers
are whole integer numbers not usually measured in fractions. For example, the
number of wings an insect may have is usually 0, 2, or 4.
s Measured numbers are just that—they have been generated by some sort of
measuring device such as a thermometer or a balance. As such they are
inexact because a certain amount of estimation is made concerning the true
value of the measurement. Measured numbers are also obtained as a result of a
mathematical operation, such as finding the mean (or “average”). The mean
value of the numbers 3, 4, 5, 5 is 4.25; the numbers are exact (assuming they
are not measurements) because they are integers but the mean is measured
because it was obtained from a calculation.
When you deal with a measured number, how do you know how accurate it
is? Let’s say you measure the length of some object and it is 5.2 centimeters
long. What does this mean? It means that it is greater than 5.1 centimeters
long but less than 5.3 centimeters long. Could it be 5.19 centimeters in length?
Yes, because of the inherent problems with measuring something accurately.
What if we say the object is 5.23 centimeters long? This means that it is
greater than 5.22 centimeters in length but less than 5.24 centimeters in
length. By adding the last digit we significantly add to the precision of the
reporting of the measurement.
And that is the point behind significant figures. They tell you how precise a
measurement is (but not necessarily how accurate). If you report something as
having a mass of 6 grams, that is different from saying that something has a
mass of 6.0 grams. The second number is more precise. The last digit gives
you a clue to the precision.
s In all measurements the last digit is always an estimate. Someone reading
the number should realize that. If you multiply two numbers, such as 3.4 and
7.319, how do you know the significance of the result? How do you know
which of the digits in a number really have value—which are significant? The
answer is in a set of rules that have been developed. These rules are as
follows:
All non-zero digits (1 through 9) are significant. [In the number 314.23
there are five significant digits.]
u
Review of Basic Mathematical Concepts
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Page 15
All zeros between non-zero digits are significant. [In the number
406.305 there are six significant digits.]
u
Zeros at the end of a number are significant if they are to the right of a
decimal point. [In the number 19.30 the last zero is significant; there are
four significant digits.] (Note that in some numbers the zeros at the end of
a number can be misleading. In the number 800, for example, are the zeros
significant? If the number is accurate, the zeros are significant. If the
number is an estimate they are not significant. This ambiguity can often be
dealt with by using scientific notation because only significant figures
should be used in scientific notation.)
u
Zeros at the beginning of a number, usually used as a placeholder, are
not significant. [Therefore, the zeros in 003 and 0.85 are not significant.]
Likewise, any zero to the left of the first significant digit is not significant.
[In 0.0045 there are only two significant digits, the 4 and 5; in scientific
notation this would be written as 4.5 ¥ 10–3.]
u
s Knowing the number of significant digits is important for being able to
report the answers of calculations. The number of significant digits reported
for any calculation cannot be any more than the fewest number of significant
digits in any of the original measured numbers used in the calculation.
For example, if you add the measured numbers 3.6 plus 0.61 plus 1.024 the
result is 5.234. However, since the fewest significant digits in any of the
original numbers was two the calculated number should be reported as 5.2.
(Rounding may be necessary, and this will be taken up in the next section.)
The same is true for subtraction. [It should be stated that not everyone agrees
with this practice. For some, the answer 5.234 is valid. To a certain extent this
may be a difference between how they are handled in science or in
mathematics. Check with your instructor as to the preferred manner of dealing
with this.]
If you multiply 26.364 times 2.53 the calculated answer is 66.70092 but since
the fewest significant digits in the original numbers is only three the number
should be reported as 66.7. This rule is generally accepted. Division is
handled in the same way.
¥
8.1
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How many significant digits are in each of the following numbers?
8.1a
8.1b
8.1c
234.765
123.057
0.0148
8.1d
8.1e
8.1f
12.430
10.00017
745.0028
Review of Basic Mathematical Concepts
Version prepared 8/23/03; for individual review purposes only
9. ROUNDING
s In doing mathematical operations, particularly using a calculator or a
computer spreadsheet, it is easy to assume that the long numbers that may
result are very precise. As seen above, this is not necessarily the case. Only a
certain amount of precision, or significance, can be given to the digits of a
calculated result. Since the last digit of most measured and calculated numbers
dealt with in a laboratory is an approximation only, it sometimes becomes
necessary to round off or to remove the additional digits the number to ensure
accuracy and consistency.
As with most things, not everyone agrees on a process for rounding. Some
techniques have been shown to be more accurate while others tend to be more
convenient. Which is used depends on a number of factors. For this laboratory
simply use the technique preferred by your lab instructor.
s Why do you round off a number? There are several answers. As seen
above, you round off a number when dealing with significant digits. You may
also round off a number for a variety of other reasons, not the least of which is
convenience or when you don’t need an exact answer.
s When do you round off a number? If you are going to round off in a
calculation always round off a number at the end of a calculation. Do not
round off any numbers within a calculation.
s How do you round off a number? Again, there are different ways of
accomplishing this.
First, you need to know how precise the rounded number needs to be. Do you
round to the nearest whole number? To the nearest million? To the nearest
thousand, hundred, ten? To the nearest tenth (one decimal place), hundredth
(two decimal places), thousandth (three decimal places)? To one significant
digit? To two or three significant digits?
The answer is usually up to you unless your lab instructor gives you explicit
directions. (“Always round to the nearest _____.”) Be consistent.
Second, you need to know how to round. The technique will vary depending
upon which precision level you choose, but, in general, the technique you
were probably taught in previous classes is generally acceptable. If the first
digit beyond the one you are rounding to is 4 or less (0, 1, 2, 3, 4) round down
to the next lowest value. If the first digit beyond the one you are rounding to is
5 or above (5, 6, 7, 8, 9) round up to the next highest value.
A more accurate way of rounding uses the following rules:
(It is assumed in the examples that you are rounding to the nearest tenth.)
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If the digit beyond the one to be retained is less than 5 (0, 1, 2, 3, 4)
don’t change the digit to be retained. Therefore, 6.43 becomes 6.4.
u
If the digit beyond the one to be retained is greater than 5 (6, 7, 8, 9)
change the digit to be retained to the next higher digit. Therefore, 6.47
becomes 6.5.
u
If the digit beyond the one to be retained is exactly 5 then look to the
digit just to the left of it. If that digit is even don’t change the digit to be
retained. If the digit is odd then change the retained digit to the next higher
digit. Therefore, 6.45 becomes 6.4 and 6.75 becomes 6.8.
u
If there are two or more digits to the right of the digit to be retained then
treat them as a group. Therefore, in 6.4321 the “321” is treated as a group,
is considered to be less than 5 (or 500), and so the number is rounded to
6.4. In 6.4652, the “652” group is evaluated as being greater than 5 (or
500) so the rounded number is 6.5.
u
¥
¥
¥
9.1
9.2
9.3
Round the following numbers to the nearest whole number.
9.1a
9.1b
9.1c
1.039
57.65
83.51
Round the following numbers to the nearest tenth.
9.2a
9.2b
9.2c
46.839
82.101
975.537
Round the following numbers to the nearest hundredth.
9.3a
9.3b
9.3c
39.4928
72.88719
0.0115
10. GRAPHING
s It might be assumed that with spreadsheets and other graphing applications
available that you would not need to know the fundamentals of graphing. This
is not true for several reasons, not the least of which is that you need to tell the
application what to graph and how to label the graph and that, in lab, no
graphing application may be available so any graphs will have to be drawn by
hand.
s The numbers you are plotting are called variables. In science there are two
types of variables, independent variables and dependent variables.
Independent variables are not changed by the experimental conditions during
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Review of Basic Mathematical Concepts
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the experiment and they do not depend on anything else. The dependent
variable (also known as the responding variable) reacts to, or depends upon,
the independent variable. Therefore, at a certain time or in a certain grouping
(independent variables) you may have a certain number of results (dependent
variable) depending upon the conditions present.
Independent variables are usually graphed along the horizontal, or X, axis of
the graph with lower values towards the left. Dependent variables are graphed
along the vertical, or Y, axis of the graph with lower values towards the
bottom. (A third, or Z, axis may be used in some cases as for plotting threedimensional data, but that will not be covered here.) This is shown below:
It is necessary that the axes themselves be marked off in appropriate intervals.
What the intervals are may vary depending on the data, but the intervals
should be evenly spaced and labeled. Make sure that both axes are labeled
with both what the variable is as well as what the units are, as in the following
example:
s The three types of graphs you are most likely to be using in this course are
scatter graphs, line graphs, and bar graphs. In a scatter graph only the points
are plotted. In a line graph, the plotted points are connected by lines.
(Technically, all the plotted points should be connected by a curved line, or a
curved line of best fit, but the same basic idea may be obtained by connecting
the points with straight lines. Use a straight edge to do so.) In a bar graph (an
example of which is called a histogram), grouped data are presented in the
form of a column or a rectangle. Examples of these three graph types are
shown below:
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Page 19
s Neatness counts. This means, among other things, that: every line should
be drawn using a straightedge; symbols should be clearly drawn and easily
recognizable; proper spacing should be used so that the graph is both easily
readable and accurate (in general, use the space available); do not go outside
the boundaries of the graph (plan your graph accordingly).
s In most cases data are plotted starting at the origin (0, 0) of the graph
unless there is a particular reason for not doing so.
s It is important, especially when the graphs contain data from multiple data
sets, that the data be identifiable. An identification key or legend, usually
placed on the upper right side of the graph, should be included. The legend
may include colors, different point types, or different hatching or fill patterns
to identify the data. These points and other identifiers should be easily
distinguishable from each other.
s Each graph should have a title describing what the graph is. That title is
usually centered above the top of the graph.
s The axes of each graph should be labeled to indicate both what the axis
represents and what units are being used.
¥
10.1
On the “graph paper” provided on the worksheet plot the following data as
two superimposed line graphs.
Mean Heart Rate (beats per minute) After Exercise
Time after exercise (min)
Males
160
0
153
1
134
2
121
3
101
4
!83
5
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Females
158
155
130
109
!87
!76
Review of Basic Mathematical Concepts
Version prepared 8/23/03; for individual review purposes only
11. SOME DEFINITIONS AND FORMULAS
s For reference, some of the more common terms and formulas that you will
need to know for this course are given below. More will be provided as
needed.
Area. The amount of surface covered by a geometric figure. It is reported in
square units or units2. (For example, two square meters = 2 m2.) Surface area
refers to the amount of surface covering the outside of a three-dimensional
object.
u
The area of a rectangle equals its length times its width.
u
The area of a circle equals pi times the radius squared. (Acircle = !r2)
Average. (See Mean.)
Circle. A geometric figure in which the outer boundary is always the same
distance from a central point. The diameter of a circle is the distance across
the center of a circle. The radius of a circle is the distance from the center of a
circle to the outside of the circle; it is equal to one-half the diameter.
Circumference. The total length of the outside of a geometric object.
The circumference of a rectangle or triangle is the sum of the length of
all of its sides.
u
The circumference of a circle is equal to pi times the diameter of the
circle (!d).
u
Diagonal. A line drawn which connects the opposite corners of a rectangle.
The diagonals of a rectangle are always equal in length.
Diameter. (See Circle.)
Mean ( X ). A statistical calculation in which the sum ( S) of the values in a
data set are divided by the number of values (N) in the data set.
Pi (!). The ratio of a circle’s circumference to its diameter. It is a number with
an infinite number of decimal places, but it is usually shortened to 3.14159 or
3.1416 or even just 3.14.
Radius. (See Circle.)
Rectangle. A geometric figure with four sides and each angle equal to 90°. A
square is a rectangle with four equal sides.
Square. (See Rectangle.)
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Page 21
Volume. The amount of space taken up by a three-dimensional geometric
figure. It is reported in cubic units or units3. (For example, nine cubic
centimeters = 9 cm3.) For regular solids, such as a cube, it represents a length
times a width times a height.
s Give these final problems a try.
¥
11.1
To the nearest tenth, what is the area of a circle with a diameter of 4.5 cm?
¥
11.2
In the diagram below is shown one quarter of a circle with a rectangle
inscribed within it. The radius of the circle is 10 centimeters. What is the
length of line AB, a diagonal of the rectangle? (It’s not hard, and it looks more
complicated than it really is. No advanced math is necessary, just an
understanding of the terms discussed above.)
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Review of Basic Mathematical Concepts
Version prepared 8/23/03; for individual review purposes only
Name: _______________________ Section: __________ Date: __________
“BASIC MATHEMATICAL CONCEPTS” WORKSHEET
4.1
4.1a
4.1b
4.1c
4.1d
4.1e
4.1f
5.1a
5.1b
5.1c
5.1d
5.1e
5.1f
8.1a
8.1b
8.1c
8.1d
8.1e
8.1f
9.1a
9.1b
9.1c
9.2a
9.2b
9.2c
9.3a
9.3b
9.3c
5.1
6.1
6.1
7.1
7.1
8.1
9.1
9.2
9.3
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10.1
11.1
11.1
11.2
11.2
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Review of Basic Mathematical Concepts
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